Derivation of the Double Porosity Model of a Compressible Miscible Displacement in Naturally Fractured Reservoirs

Abstract

We derive a double porosity model for a displacement of one compressible miscible fluid by another in a naturally fractured reservoir. The microscopic model consists of the usual equations describing Darcy flow in a reservoir except that the porosity and the permeability coefficients are highly discontinuous. The viscosity is assumed to be constant. Over the matrix domain, the coefficients are scaled by a parameter ϵ representing the size of the matrix blocks. This scaling preserves the physics of the flow in the matrix as ϵ tends to zero. Using homogenization theory, we derive rigorously the corresponding double porosity model. To this purpose, we mainly use the concept of two-scale convergence. The less permeable part of the rock then contributes as nonlinear memory terms. To specify them in spite of the strong nonlinearities and of the coupling, we then use some appropriate dilation operator.